More on linear transformations

In summary, a linear transformation is a mathematical function that preserves the structure of a vector space while mapping it to another. It is represented by a matrix, with the columns representing the images of the basis vectors and the rows representing the coordinates of the images. The main difference between a linear and non-linear transformation is that the former preserves linearity and proportionality, while the latter does not. Linear transformations are used in various fields, such as computer graphics and economics, to model and analyze linear systems and solve optimization problems. The inverse of a linear transformation is a function that reverses the effects of the transformation, represented by the inverse matrix of the original transformation matrix.
  • #1
johnnyboy2005
29
0
i think I'm just having a hard understanding linear transformations...

i was asked if (5, 0) is a vector in R(T) given by the formula
T(x,y)=(2x-y,-8x + 4y)...i really don't get what I'm supposed to do here.. any hints would be most appreciated.
 
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  • #2
It asks if the vector (5,0) is in the range of the function T.
So does there exist some vector (x,y), such that T(x,y)=(5,0)?

How would you go about this problem?
 
  • #3
so much easier now. thank you Galileo
 

1. What is a linear transformation?

A linear transformation is a function that maps one vector space to another, while preserving the algebraic structure of the original space. It is a type of mathematical operation that can be applied to a set of numbers or geometric objects, resulting in a new set of numbers or objects.

2. How is a linear transformation represented?

A linear transformation can be represented by a matrix. The columns of the matrix represent the images of the basis vectors of the original space, and the rows represent the coordinates of the images in the new space.

3. What is the difference between a linear transformation and a non-linear transformation?

A linear transformation preserves the proportionality and linearity of vectors, while a non-linear transformation does not. In other words, the output of a linear transformation is always a straight line or a plane, while the output of a non-linear transformation can be curved or distorted.

4. How are linear transformations used in real life?

Linear transformations are used in various fields of science and engineering, such as computer graphics, physics, and economics. They are used to model and analyze linear systems, such as electrical circuits and mechanical systems, and to solve optimization problems.

5. What is the inverse of a linear transformation?

The inverse of a linear transformation is a function that undoes the transformation by mapping the output back to the original input. It is represented by the inverse matrix of the original transformation matrix, and it allows for the reversal of the effects of the transformation.

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